Relaxation of isolated quantum systems beyond chaos

In classical statistical mechanics there is a clear correlation between relaxation to equilibrium and chaos. In contrast, for isolated quantum systems this relation is—to say the least—fuzzy. In this work we try to unveil the intricate relation between the relaxation process and the transition from integrability to chaos. We study the approach to equilibrium in two different many-body quantum systems that can be parametrically tuned from regular to chaotic. We show that a universal relation between relaxation and delocalization of the initial state in the perturbed basis can be established regardless of the chaotic nature of system.

Figure 1

Top: $S_{\mathrm{dec}}-\overline{S_{\mathrm{D}}\left(\tau \right)}$. Bottom: Averaged fluctuations of $S_{\mathrm{D}}\left(\tau \right)$ as a function of ${\lambda }_0$, with $\delta \lambda =0.1$. The vertical (solid) line marks the transition to chaotic behavior at ${\lambda }_{\mathrm{c}}=0.5$. The shaded area marks the region where at least one Hamiltonian $[H\left({\lambda }_0\right)$ or $H({\lambda }_0+\delta \lambda )]$ corresponds to chaotic dynamics. The horizontal dashed (top panel) line marks universal value $1-\gamma \approx 0.423$. The symbols correspond to different initial states $|n\rangle$ ($H|n\rangle =E_n|n\rangle$): $\diamond ,|10\rangle$; $▿,|100\rangle$; $▵,|500\rangle$; $\square ,|1000\rangle$; $\circ ,|2000\rangle$. Inset: The dashed green line on top marks $S_{\mathrm{dec}}$ while the dashed gray line marks $S_{\mathrm{dec}}-1+\gamma$. DM with $J=20,\lambda =0.65$ ($n_{\max }=250$) energy $E_{501}$. The arrows indicate the distance $S_{\mathrm{dec}}-S_{\mathrm{D}}\left(\tau \right)\approx 1-\gamma =0.4228...$ .

Figure 2

$S_{\mathrm{dec}}-\overline{S_{\mathrm{D}}\left(\tau \right)}$ as a function of $\xi$. Main panel: DM with $j=20,\lambda \le 1.0$ and $\delta \lambda =0.1$, and $n_{\max }=250$. The color box indicates the values of $\lambda$. The dashed, horizontal line marks the value $1-\gamma =0.4228...$. The symbols correspond to different initial states: $\diamond ,|10\rangle$; $▿,|100\rangle$; $▵,|500\rangle$; $\square ,|1000\rangle$; $\circ ,|2000\rangle$; light-gray symbols correspond to results for $j=10$. The solid blue curve is the approximating function $(1-\gamma )(\xi -1)/(\xi +1)$. Inset: The white $\circ$'s correspond to results for the spin model with $\mu =0.5$. The gray symbols correspond to the DM results shown in the main panel (light gray, $j=10$; dark gray, $j=20$).

Figure 3

Main: Fluctuations $\Delta S_{\mathrm{D}}/\langle \overline{S_{\mathrm{D}}\left(\tau \right)}\rangle$ as a function of $\xi$ for the DM with same parameter values as Fig. 2. The color box indicates the values of ${\lambda }_0$. The dashed blue line is $1/\xi$. Inset: Fluctuations for the spin model ($\circ$); the gray symbols correspond to the DM results (light gray, $j=10$; dark gray, $j=20$).

Figure 4

$\xi$ (IPR) as a function of $\delta \lambda$ for the DM with $j=20$ ($n_{\max }=200$), initial state $|100\rangle$, and different values of ${\lambda }_0$. In the inset we show the same for the spin model with $\mu =0.5$. In both cases the values of ${\lambda }_0$ are $\square$, 0.01; $+$, 0.1; $\blacksquare$, 0.2; $▴$, 0.3; $\circ$, 0.4; $•$, 0.5; $▵$, 0.7; and $\diamond$, 1.0. The light-gray symbols in the inset correspond to the ones in the main panel (“eye guide”).